What is the square root of 523,457?
If your calculator doesn't have a square root button, finding the square root of a large number can be rather difficult and time consuming. The square root of most whole numbers is an irrational number — that is, an infinitely long, non-repeating decimal. If the number you're dealing with isn't a perfect square (like 9 or 25), then the best you can do is approximate the value of the square root by determining which two numbers it lies between.
Perfect squares never end in 7 (or 2, 3, or 8; think about how you multiply numbers and you'll see why), so you know that 523,457 isn't a perfect square. You need to estimate the square based on the perfect squares that lie on either side of 523,457.
Start by looking at the nearby perfect squares that you can figure in your head. In this case, 490,000 (7002) and 640,000 (8002) work well. Now you know that the answer is a three-digit number in the seven hundreds.
Next, figure out the second digit. Because 523,457 is closer to 490,000 than to 640,000, its square root will be closer to 700 than to 800, so the second digit is less than 5. Try 740:
7402 = 547,600
This is still a bit too high, so keep trying lower numbers in that second spot until you hit one that's too small:
7302 = 532,900
7202 = 518,400
Aha! So now you know that the square root of 523,457 lies somewhere between 720 and 730. Because 523,457 is closer to 518,400 than to 532,900, its square root will be closer to 720 than to 730. That last digit must be less than 5!
Now you can really start to narrow it down:
7242 = 524,176
7232 = 522,729
And there you have it! 523,457 is about halfway between 524,176 and 522,729, so you can approximate the square root of 523,457 as 723.5.
You can continue this narrowing-down process infinitely into the decimal domain, or at least to as many decimal places as you want or need.